Question: Multiply the following complex numbers, marked as blue dots on the graph: $[4(\cos(\frac{5}{6}\pi) + i \sin(\frac{5}{6}\pi))] \cdot [\cos(\frac{5}{12}\pi) + i \sin(\frac{5}{12}\pi)]$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $4(\cos(\frac{5}{6}\pi) + i \sin(\frac{5}{6}\pi))$ ) has angle $\frac{5}{6}\pi$ and radius $4$ The second number ( $\cos(\frac{5}{12}\pi) + i \sin(\frac{5}{12}\pi)$ ) has angle $\frac{5}{12}\pi$ and radius $1$ The radius of the result will be $4 \cdot 1$ , which is $4$ The angle of the result is $\frac{5}{6}\pi + \frac{5}{12}\pi = \frac{5}{4}\pi$ The radius of the result is $4$ and the angle of the result is $\frac{5}{4}\pi$.